29 problems

Let $K$ be a knot, and let $V_K(x)$ denote its Jones polynomial in the normalization used here. Jones polynomial unknot-detection conjecture. … The conjecture asks whether the Jone…

Let $b$ be an integer, and consider quasi-alternating links whose Jones polynomial has breadth $b$, where the breadth is the difference between the highest and lowest exponents occ…

Suppose that $L$ is an oriented link in $S^3$ with $mu$ components. Write its averaged Jones polynomial at $t=1$ as … and set $\phi_n(L)=(-2)^{\mu}a_{n+\mu}(L)$. An algebraically s…

Let $K$ be a knot, and let $V_K(t)$ denote its Jones polynomial. A knot is nontrivial if it is not equivalent to the unknot. Jones polynomial detection conjecture. Every nontrivial…

Asymptotic Vassiliev-invariant conjecture. The numbers $\log_{v_2(K_i)}v_3(K_i)$, which are well-defined for $K_i\ne !3_1$, converge to $2$ as $i\to\infty$. This predicts a precise…

Let $K$ be a hyperbolic knot, let $J_K$ denote its Jones polynomial, and let the volume of its complement be $\operatorname{vol}(S^3{\smallsetminus} K)$. Volume conjecture. The vol…

Let $L$ be a prime knot. Write ${\text{\it Kh}}_{\mathbb Q}(L)$ and ${\text{\it Kh}}_{{\mathbb F}_2}(L)$ for its rational and mod-2 Khovanov polynomials, where ${\mathbb F}_2$ is t…

Let $B_n$ be the braid group on $n$ strands, with standard generators $sigma_1,sigma_2,dots,sigma_{n-1}$. For a braid, its closure is the link obtained by joining corresponding end…

Let $B_n$ be a braid group, and let its Temperley–Lieb representation be the representation into the Temperley–Lieb algebra. A knot is non-trivial if it is not isotopic to the unkn…

Let $L$ be a quasi-alternating link, let $V_L(t)$ denote its Jones polynomial, and let $\operatorname{breadth}(V_L)$ be the difference between the highest and lowest exponents of…

Odd-degree Jones coefficient nonvanishing conjecture. For every $k\geq 1$,

The primitivity and minimality conjecture. For all $k>1$, $\lambda_{2k}w_{2k}$ is a primitive integer-valued Vassiliev invariant, and $\lambda_{2k}$ is minimal. The authors also co…

The Alexander polynomial can be defined using a method known as colouring. The Jones-polynomial colouring problem. Can the Jones polynomial be defined via colouring? The source pre…

A rational knot is represented by an integer sequence presentation, and a Jones rational coincidence is a pair of rational knots with the same Jones polynomial. The pivoting-pairs…

Chbili–Qazaqzeh's conjecture. The coefficients satisfy

Let $L$ be a link whose Jones polynomial is written as … where $m\geq 0$, $a_0\neq 0$, and $a_m\neq 0$. Chbili–Qazaqzeh conjecture. If $L$ is a prime quasi-alternating link, other…

Let the Jones polynomial of $W(3,n)$ have the form … For the coefficients indexed by $k$, write $\lambda_{-n+k}$ and $\lambda_{n-k}$ for the corresponding coefficients. Polynomial-…

Let $W(3,n)$ be a weaving knot, and write $T_k(W(3,n))$ for its $k$th twist number. Polynomial-degree conjecture. The function $T_k(W(3,n))$ is a polynomial in $n$ of degree $k$. T…

Let $K$ be a knot, and let $V_K(t)$ denote its Jones polynomial. The trivial knot is the knot with no crossings, while a non-trivial knot is any knot not equivalent to it. Jones co…

Non-alternating maximality conjecture. The alternating hypothesis in the corresponding proposition can be removed: every sequence of diagrammatically maximal links satisfies

Breadth–determinant conjecture. If $L$ is a quasi-alternating link, then

Condensed three-braid conjecture. One has

A virtual knot is Z-move equivalent to the unknot if it can be reduced to the unknot by a series of Reidemeister moves for virtual knots and Z-moves. Z-move conjecture. Every knot…

Let $\kappa=\kappa_\beta$ be the closure of a braid $\beta\in Br_m$ with writhe $w$, and let $P_{\mathrm{coker}}(t)$ denote the associated cokernel polynomial. Cokernel polynomial…

Let a knot be given by a plat presentation of linear length, and let $b$ denote its bridge number. Evaluate the Jones polynomial at a dense root of unity and consider the running t…